A Birthday Paradox for Markov chains with an optimal bound for collision in the Pollard Rho algorithm for discrete logarithm

Abstract

We show a Birthday Paradox for self-intersections of Markov chains with uniform stationary distribution. As an application, we analyze Pollard's Rho algorithm for finding the discrete logarithm in a cyclic group G and find that if the partition in the algorithm is given by a random oracle, then with high probability a collision occurs in Θ( √ |G|) steps. Moreover, for the parallelized distinguished points algorithm on J processors we find that Θ( √ |G|/J ) steps suffices. These are the first proofs of the correct order bounds which do not assume that every step of the algorithm produces an i.i.d. sample from G. © Institute of Mathematical Statistics, 2010.